It's supposed to be an efficiency measure. i.e. when you're taking a substring you won't create a new char array, but merely create a window onto the existing char array.

Is this worthwhile ? Maybe. The downside is that it causes some confusion (e.g. see this SO question), plus each String object needs to carry the offset info into the array, even if it's not used.

EDIT: This behaviour has now changed as of Java 7. See the linked answer for more info

I'm assuming the problems you're having with indexOf() related to you using the character version (otherwise why would you be searching for w when looking for world?). If so, indexOf() can take a string argument to search for:

String s = "hello world i am from heaven";
if (s.indexOf("world") != -1) {
  // it contains world
}

as for log base 2, that's easy:

public static double log2(double d) {
  return Math.log(d) / Math.log(2.0d);
}

String example = "/abc/def/ghfj.doc";
System.out.println(example.substring(example.lastIndexOf("/") + 1));